Lemma 3 ([1]})
Given a matrix \(M \in \mathbb {R}^{m \times n}\) , let \(P^{\prime } = \lbrace P^{\prime }_{1}, P^{\prime }_{2}, \cdots , P^{\prime }_{m}\rbrace \)  be a probability distribution on \([m]\)  such that \(P^{\prime }_{i} = \frac{\Vert M_{i,:}\Vert ^2}{\Vert M\Vert ^2_F}, i \in [m]\) . Let \((i_1, i_2, \cdots , i_p)\)  be a sequence of \(p\)  independent samples from \([m]\) , each chosen according to distribution \(P^{\prime }\) . Let \(N\)  be the \({p \times n}\)  matrix with
\(N_{t,:} = \frac{M_{i_t,:}}{\sqrt{pP^{\prime }_{i_t}}}, \, t \in [p].\) 
